Discrte Topology and Trivial topology: An adjoint functor point of view.

Consider the forget functor $F:\mathrm{Top}\to \mathrm{Set}$$F:\mathrm{Top}\to\mathrm{Set}$.

It has both left and right adjoints.

The left adjoint $U:\mathrm{S}\mathrm{e}\mathrm{t}\to \mathrm{T}\mathrm{o}\mathrm{p}$$U:\rm Set\to Top$

$$\begin{array}{}\text{(1)}& X⟼(X,\tau ),\tau =P(X)\end{array}$$

Let us prove that $U$$U$ is left adjoint of $F$$F$.

That is

$$\begin{array}{}\text{(2)}& {\mathrm{Hom}}_{\mathrm{Top}}(U(X),Y)\cong {\mathrm{Hom}}_{\mathrm{Set}}(X,F(Y))\end{array}$$

Proof.

It is just a translation of all the functions from $U(X)\to Y$$U(X)\to Y$ is continuous.

The right adjoint $V:\mathrm{S}\mathrm{e}\mathrm{t}\to \mathrm{T}\mathrm{o}\mathrm{p}$$V:\rm Set\to Top$.

$$\begin{array}{}\text{(3)}& Y⟼(Y,{\tau }^{\prime }),{\tau }^{\prime }=\{\mathrm{\varnothing },X\}\end{array}$$

That is

 

$$\begin{array}{}\text{(4)}& {\mathrm{Hom}}_{\mathrm{Set}}(F(X),Y)\cong {\mathrm{Hom}}_{\mathrm{Top}}(X,V(Y))\end{array}$$

Proof.

It is a translation of all the functions from $X\to V(Y)$$X\to V(Y)$ is continuous.